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We consider light scattering from ultracold atoms trapped in an optical lattice into a cavity. The measurement of photons leaking out the cavity provides a quantum nondemolition (QND) access to various atomic variables. Depending on the chosen geometry the time resolved light detection projects the atomic quantum state to various atom-number squeezed and Schroedinger cat states. Modifications of the atomic and light properties at a single quantum trajectory are demonstrated. The quantumdoi:10.1103/physrevlett.102.020403 pmid:19257251 fatcat:cziunhtvwzbnxit7zbmonfmdkm